The maximal success probability in Eq. (REF ) is called the Helstrom bound[1]}, which is analytically given by,
\(P_{hel}(|\psi _0\rangle ,|\psi _1\rangle |q_0,q_1)&:=&\max _{\lbrace M_0,M_1\rbrace }P_s(|\psi _0\rangle ,|\psi _1\rangle |q_0,q_1)\nonumber \\&=&\frac{1+\sqrt{1-4q_0q_1|\langle \psi _0|\psi _1\rangle |^2}}{2}.\) 
